A230119 Numbers of quasi-pyramid polycubes of a given volume (number of atomic cells).

3, 9, 23, 47, 91, 169, 291, 494, 815, 1295, 2043, 3155, 4775, 7165, 10599, 15458, 22455, 32145, 45659, 64519, 90247, 125493, 173515, 238153, 325423, 442169, 597575, 804203, 1077283, 1436593, 1908571, 2525611, 3329391, 4373969, 5726611, 7472763, 9721983, 12608475, 16305179, 21027921, 27043631, 34689187, 44385995, 56652421, 72137483, 91645813, 116167379, 146932651, 185456419, 233594265

Offset: 1

Author

Matthieu Deneufchâtel, Oct 10 2013

Keywords

• nonn

Comments

A pyramid polycube is obtained by gluing together horizontal plateaux (parallelepipeds of height 1) in such a way that (0,0,0) belongs to the first plateau and each cell with coordinate (0,b,c) belonging to the first plateau is such that b , c >= 0. If the cell with coordinates (a,b,c) belongs to the (a+1)-st plateau (a>0), then the cell with coordinates (a-1, b, c) belongs to the a-th plateau.

A quasi-pyramid polycube is an object obtained from a pyramid by removing all the cells with coordinates (a,b,c) with 1 <= a <= h for a fixed pair (b,c) chosen among the triples (h,b,c) such that there is a cell with coordinates (h,b,c) in the pyramid ((h,b,c) belongs to the highest plateau of the pyramid).

If Q(x) denotes the generating function of the quasi-pyramids and P(x,h) the generating function of the pyramids counted by height, then the x^(-h) P(x,h) converges when h goes to infinity and the limit is Q(x) + x/(1-x).

Crossrefs

A229914, A227926

Programs

Maple

calcPyr:=proc(i, j, k, l) option remember;
if (l<0) then 0
elif (i*j*k>l) then 0
elif k=1 then if (i*j=l) then 1 else 0; fi;
else s:=0; a:=0; b:=0;
while ((i+a)*j*(k-1)<=l-i*j) do
b:=0;
while ((i+a)*(j+b)*(k-1)<=l-i*j) do
s:=s+(a+1)*(b+1)*calcPyr(i+a, j+b, k-1, l-i*j);
b:=b+1;
od;
a:=a+1;
od;
s;
fi;
end;
countPyr:=proc(l)
s:=0;
for k to l do
i:=1:
while (i*k<=l) do
j:=1;
while (i*k*j<=l) do
s:=s+t^k*calcRecPyr(i, j, k, l);
j:=j+1;
od:
i:=i+1;
od;
od;
s;
end;
enum=[seq(countPyr(ii), ii=1..200)];
serie_quasiPyr:=proc(l) global enum;local ii;
map(coeff,enump,t^l);
select(x->x>0,%);
sum(t^(ii-1)*%[ii],ii=1..nops(%));
end;
serie_quasiPyr(100):
[1,seq(coeff(%,t^ii)-1,ii=1..50)];